Thm: Given any , there exists exactly one solution such that or . Proof: Given , it is either even or odd. If is even then is a solution. If is odd, then is even, so is a solution. Call this initial solution . (Note that if is odd but , no positive solution exists, not even this initial one.)

Cor: Now for any , is also a solution. To be a strictly positive solution, , so for a second solution to exist.

Thus, for a second solution to NOT exist, or , and . So, consider the function on the domain . These resulting values of are the ones for which only one positive solution exists.